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Theorem dvlog2lem 20533
Description: Lemma for dvlog2 20534. (Contributed by Mario Carneiro, 1-Mar-2015.)
Hypothesis
Ref Expression
dvlog2.s  |-  S  =  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )
Assertion
Ref Expression
dvlog2lem  |-  S  C_  ( CC  \  (  -oo (,] 0 ) )

Proof of Theorem dvlog2lem
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dvlog2.s . . . . 5  |-  S  =  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )
2 cnxmet 18797 . . . . . 6  |-  ( abs 
o.  -  )  e.  ( * Met `  CC )
3 ax-1cn 9038 . . . . . 6  |-  1  e.  CC
4 1re 9080 . . . . . . 7  |-  1  e.  RR
54rexri 9127 . . . . . 6  |-  1  e.  RR*
6 blssm 18438 . . . . . 6  |-  ( ( ( abs  o.  -  )  e.  ( * Met `  CC )  /\  1  e.  CC  /\  1  e.  RR* )  ->  (
1 ( ball `  ( abs  o.  -  ) ) 1 )  C_  CC )
72, 3, 5, 6mp3an 1279 . . . . 5  |-  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )  C_  CC
81, 7eqsstri 3370 . . . 4  |-  S  C_  CC
98sseli 3336 . . 3  |-  ( x  e.  S  ->  x  e.  CC )
103subid1i 9362 . . . . . . . . 9  |-  ( 1  -  0 )  =  1
11 mnfxr 10704 . . . . . . . . . . . 12  |-  -oo  e.  RR*
12 0re 9081 . . . . . . . . . . . 12  |-  0  e.  RR
13 iocssre 10980 . . . . . . . . . . . 12  |-  ( ( 
-oo  e.  RR*  /\  0  e.  RR )  ->  (  -oo (,] 0 )  C_  RR )
1411, 12, 13mp2an 654 . . . . . . . . . . 11  |-  (  -oo (,] 0 )  C_  RR
1514sseli 3336 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  x  e.  RR )
1612a1i 11 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  0  e.  RR )
174a1i 11 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  1  e.  RR )
18 elioc2 10963 . . . . . . . . . . . 12  |-  ( ( 
-oo  e.  RR*  /\  0  e.  RR )  ->  (
x  e.  (  -oo (,] 0 )  <->  ( x  e.  RR  /\  -oo  <  x  /\  x  <_  0
) ) )
1911, 12, 18mp2an 654 . . . . . . . . . . 11  |-  ( x  e.  (  -oo (,] 0 )  <->  ( x  e.  RR  /\  -oo  <  x  /\  x  <_  0
) )
2019simp3bi 974 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  x  <_  0 )
2115, 16, 17, 20lesub2dd 9633 . . . . . . . . 9  |-  ( x  e.  (  -oo (,] 0 )  ->  (
1  -  0 )  <_  ( 1  -  x ) )
2210, 21syl5eqbrr 4238 . . . . . . . 8  |-  ( x  e.  (  -oo (,] 0 )  ->  1  <_  ( 1  -  x
) )
23 ax-resscn 9037 . . . . . . . . . . . 12  |-  RR  C_  CC
2414, 23sstri 3349 . . . . . . . . . . 11  |-  (  -oo (,] 0 )  C_  CC
2524sseli 3336 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  x  e.  CC )
26 eqid 2435 . . . . . . . . . . 11  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
2726cnmetdval 18795 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  x  e.  CC )  ->  ( 1 ( abs 
o.  -  ) x
)  =  ( abs `  ( 1  -  x
) ) )
283, 25, 27sylancr 645 . . . . . . . . 9  |-  ( x  e.  (  -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  =  ( abs `  (
1  -  x ) ) )
29 0le1 9541 . . . . . . . . . . . 12  |-  0  <_  1
3029a1i 11 . . . . . . . . . . 11  |-  ( x  e.  (  -oo (,] 0 )  ->  0  <_  1 )
3115, 16, 17, 20, 30letrd 9217 . . . . . . . . . 10  |-  ( x  e.  (  -oo (,] 0 )  ->  x  <_  1 )
3215, 17, 31abssubge0d 12224 . . . . . . . . 9  |-  ( x  e.  (  -oo (,] 0 )  ->  ( abs `  ( 1  -  x ) )  =  ( 1  -  x
) )
3328, 32eqtrd 2467 . . . . . . . 8  |-  ( x  e.  (  -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  =  ( 1  -  x ) )
3422, 33breqtrrd 4230 . . . . . . 7  |-  ( x  e.  (  -oo (,] 0 )  ->  1  <_  ( 1 ( abs 
o.  -  ) x
) )
35 cnmet 18796 . . . . . . . . . 10  |-  ( abs 
o.  -  )  e.  ( Met `  CC )
3635a1i 11 . . . . . . . . 9  |-  ( x  e.  (  -oo (,] 0 )  ->  ( abs  o.  -  )  e.  ( Met `  CC ) )
373a1i 11 . . . . . . . . 9  |-  ( x  e.  (  -oo (,] 0 )  ->  1  e.  CC )
38 metcl 18352 . . . . . . . . 9  |-  ( ( ( abs  o.  -  )  e.  ( Met `  CC )  /\  1  e.  CC  /\  x  e.  CC )  ->  (
1 ( abs  o.  -  ) x )  e.  RR )
3936, 37, 25, 38syl3anc 1184 . . . . . . . 8  |-  ( x  e.  (  -oo (,] 0 )  ->  (
1 ( abs  o.  -  ) x )  e.  RR )
40 lenlt 9144 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  ( 1 ( abs 
o.  -  ) x
)  e.  RR )  ->  ( 1  <_ 
( 1 ( abs 
o.  -  ) x
)  <->  -.  ( 1 ( abs  o.  -  ) x )  <  1 ) )
414, 39, 40sylancr 645 . . . . . . 7  |-  ( x  e.  (  -oo (,] 0 )  ->  (
1  <_  ( 1 ( abs  o.  -  ) x )  <->  -.  (
1 ( abs  o.  -  ) x )  <  1 ) )
4234, 41mpbid 202 . . . . . 6  |-  ( x  e.  (  -oo (,] 0 )  ->  -.  ( 1 ( abs 
o.  -  ) x
)  <  1 )
432a1i 11 . . . . . . 7  |-  ( x  e.  (  -oo (,] 0 )  ->  ( abs  o.  -  )  e.  ( * Met `  CC ) )
445a1i 11 . . . . . . 7  |-  ( x  e.  (  -oo (,] 0 )  ->  1  e.  RR* )
45 elbl2 18410 . . . . . . 7  |-  ( ( ( ( abs  o.  -  )  e.  ( * Met `  CC )  /\  1  e.  RR* )  /\  ( 1  e.  CC  /\  x  e.  CC ) )  -> 
( x  e.  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )  <->  ( 1 ( abs  o.  -  ) x )  <  1 ) )
4643, 44, 37, 25, 45syl22anc 1185 . . . . . 6  |-  ( x  e.  (  -oo (,] 0 )  ->  (
x  e.  ( 1 ( ball `  ( abs  o.  -  ) ) 1 )  <->  ( 1 ( abs  o.  -  ) x )  <  1 ) )
4742, 46mtbird 293 . . . . 5  |-  ( x  e.  (  -oo (,] 0 )  ->  -.  x  e.  ( 1 ( ball `  ( abs  o.  -  ) ) 1 ) )
4847con2i 114 . . . 4  |-  ( x  e.  ( 1 (
ball `  ( abs  o. 
-  ) ) 1 )  ->  -.  x  e.  (  -oo (,] 0
) )
4948, 1eleq2s 2527 . . 3  |-  ( x  e.  S  ->  -.  x  e.  (  -oo (,] 0 ) )
509, 49eldifd 3323 . 2  |-  ( x  e.  S  ->  x  e.  ( CC  \  (  -oo (,] 0 ) ) )
5150ssriv 3344 1  |-  S  C_  ( CC  \  (  -oo (,] 0 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725    \ cdif 3309    C_ wss 3312   class class class wbr 4204    o. ccom 4874   ` cfv 5446  (class class class)co 6073   CCcc 8978   RRcr 8979   0cc0 8980   1c1 8981    -oocmnf 9108   RR*cxr 9109    < clt 9110    <_ cle 9111    - cmin 9281   (,]cioc 10907   abscabs 12029   * Metcxmt 16676   Metcme 16677   ballcbl 16678
This theorem is referenced by:  dvlog2  20534  logtayl  20541  logtayl2  20543  efrlim  20798  lgamcvg2  24829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-n0 10212  df-z 10273  df-uz 10479  df-rp 10603  df-xadd 10701  df-ioc 10911  df-seq 11314  df-exp 11373  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-psmet 16684  df-xmet 16685  df-met 16686  df-bl 16687
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